Past Junior Topology and Group Theory Seminar

Waldhausen defined higher K-groups for categories with certain extra
structure. In this talk I will define categories with cofibrations and
weak equivalences, outline Waldhausen's construction of the associated
K-Theory space, mention a few important theorems and give some examples.
If time permits I will discuss the infinite loop space structure on the
K-Theory space.

A one hour introduction to topological K-theory, that
nifty generalised cohomology theory that is built starting from the
semi-ring of vector bundles over a space. As I'll need it on Thursday I'll also explain a model for K-theory in terms of
difference bundles, and, if time permits, its connection with Clifford algebras.

Subgroup separability is a group-theoretic property that has important
implications for geometry and topology, because it allows us to lift
immersions to embeddings in a finite sheeted covering space. I will
describe how this works in the case of graphs, and go on to motivate the
construction of special cube complexes as an attempt to generalise the
technique to higher dimensions.

As the title says, in this talk I will be giving a casual introduction
to higher categories. I will begin by introducing strict n-categories
and look closely at the resulting structure for n=2. After discussing
why this turns out to be an unsatisfying definition I will discuss in
what ways it can be weakened. Broadly there are two main classes of
models for weak n-categories: algebraic and geometric. The differences
between these two classes will be demonstrated by looking at
bicategories on the algebraic side and quasicategories on the geometric.

Kazhdan's
Property (T) is a powerful property of groups, with many useful
consequences. Probably the best known examples of groups with (T) are
higher rank lattices. In this talk I will provide a proof
that for n ≥ 3, SLn(ℤ) has (T). A nice feature of the
approach I will follow is that it works entirely within the world of
discrete groups: this is in contrast to the classical method, which
relies on being able to embed a group as a lattice
in an ambient Lie group.

In some of their recent work Derbez and Wang studied volumes of representations of 3-manifold groups into the Lie groups $$Iso_e \widetilde{SL_2(\mathbb{R})} \mbox{ and }PSL(2,\mathbb{C}).$$ They computed the set of all volumes of representations for a fixed prime closed oriented 3-manifold with $$\widetilde{SL_2(\mathbb{R})}\mbox{-geometry}$$ and used this result to compute some volumes of Graph manifolds after passing to finite coverings.
In the talk I will give a brief introduction to the theory of volumes of representations and state some of Derbez' and Wang's results. Then I will prove an additivity formula for volumes of representations into $$Iso_e \widetilde{SL_2(\mathbb{R})}$$ which enables us to improve some of the results of Derbez and Wang.

I will explain why one can symplectically embed closed symplectic
manifolds (with integral symplectic form) into CPn and compute the
weak homotopy type of the space of all symplectic embeddings of such a
symplectic manifold into CP∞.

The notion of automatic groups emerged from conversations between Bill
Thurston and Jim Cannon on the nice algorithmic properties of Kleinian
groups. In this introductory talk we will define automatic groups and
then discuss why they are interesting. This centres on how automatic
groups subsume many other classes of groups (e.g. hyperbolic groups,
finitely generated Coxeter groups, and braid groups) and have good
properties (e.g. finite presentability, fast solution to the word
problem, and type FP∞).